The influence of data regularity in the critical exponent for a class of semilinear evolutions equations

TL;DR

This paper determines the critical exponent for the global existence of small data solutions to a class of semilinear dissipative evolution equations, highlighting how data regularity influences the threshold for solution blow-up or persistence.

Contribution

It establishes the critical exponent for global solutions considering data regularity and extends understanding of semilinear dissipative equations with fractional operators.

Findings

Critical exponent p_c=1+ (2mθ)/n for global existence

Nonexistence of solutions in subcritical cases proved

Regularity conditions influence the critical exponent

Abstract

In this paper we find the critical exponent for the global existence (in time) of small data solutions to the Cauchy problem for the semilinear dissipative evolution equations % \[ u_{tt}+(-\Delta)^\delta u_{tt}+(-\Delta)^\alpha u+(-\Delta)^\theta u_t=|u_t|^p, \quad t\geq 0,\,\, x\in\R^n,\] % with $p>1$, $2\theta \in [0, \alpha]$ and $\delta \in (\theta,\alpha]$. We show that, under additional regularity $\left(H^{\alpha+\delta}(\R^n)\cap L^{m}(\R^n) \right)\times \left(H^{2\delta}(\R^n)\cap L^{m}(\R^n)\right) $ for initial data, with $m\in (1,2]$, the critical exponent is given by $p_c=1+\frac{2m\theta}{n}$. The nonexistence of global solutions in the subcritical cases is proved, in the case of integers parameters $\alpha, \delta, \theta$, by using the test function method (under suitable sign assumptions on the initial data).